△ABC is an isosceles triangle with legs AB and AC. △AYX is also an isosceles triangle with legs AY and AX. Triangle A Y X is shown. Line segment B C is drawn from side A Y to A X to form triangle A B C. The proof that △ABC ~ △AYX is shown. Statements Reasons 1. △ABC is isosceles with legs AB and AC; △AYX is also isosceles with legs AY and AX. 1. given 2. AB ≅ AC and AY ≅ AX 2. definition of isosceles triangle 3. AB = AC and AY = AX 3. definition of congruency 4. AY • AC = AX • AC 4. multiplication property of equality 5. AY • AC = AX • AB 5. substitution property of equality 6. 6. division property of equality 7. 7. division property of equality 8. ? 8. ? 9. △ABC ~ △AYX 9. SAS similarity theorem Which statement and reason are missing in the proof?

Explanation:

Here is our take on the proof shown in the problem statement. The missing statements and reasons are shown in bold.

Statements  Reasons

1. △ABC is isosceles with legs AB and AC; △AYX is also isosceles with legs AY and AX. 1. given

2. AB ≅ AC and AY ≅ AX  2. definition of isosceles triangle

3. AB = AC and AY = AX  3. definition of congruency

4. AY • AC = AX • AC  4. multiplication property of equality

5. AY • AC = AX • AB  5. substitution property of equality

6. AY • AC/AX = AB  6. division property of equality

7. AC/AX = AB/AY  7. division property of equality

8. Corresponding sides are proportional  8. Definition of proportional

9. △ABC ~ △AYX  9. SAS similarity theorem

The reason given in statements 6 and 7 suggest you need to divide something. For SAS similarity, you need to show corresponding sides are proportional. The missing steps are to get to the point where you can say that.

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